"How do you find the equation of the..." was answered by our experts

Vanessa Mccarty

Answered question

2022-04-15

How do you find the equation of the tangent line to the curve

f (x) = (\sec x) (\tan x)

x = \frac{π}{6}

Answer & Explanation

Austin Sherman

Beginner2022-04-16Added 12 answers

To find tangent line, we need to find the point at which tangent is drawn and its slope and then use point slope form of equation.
Former is easily available from f(x) as

f (\frac{π}{6}) = \sec (\frac{π}{6}) \tan (\frac{π}{6}) = \frac{2}{\sqrt{3}} \times \frac{1}{\sqrt{3}} = \frac{2}{3}

Hence tangent is desired at

(\frac{π}{6}, \frac{2}{3})

.
For slope we need

\frac{d f}{d x} = \sec x \times \sec^{2} x + \tan x \times \sec x \tan x

= \sec^{3} x + \sec x \tan^{2} x

and at

x = \frac{π}{6}

, it is

{(\frac{2}{\sqrt{3}})}^{3} + \frac{2}{\sqrt{3}} \times {(\frac{1}{\sqrt{3}})}^{2} = \frac{8}{3 \sqrt{3}} + \frac{2}{3 \sqrt{3}} = \frac{10}{3 \sqrt{3}}

As equation of a line of slope m passing through

(x_{1}, y_{1})

(y - y_{1}) = m (x - x_{1})

and hence equation of tangent is

(y - \frac{2}{3}) = \frac{10}{3 \sqrt{3}} (x - \frac{π}{6})

3 \sqrt{3} y - 2 \sqrt{3} = 10 (x - \frac{π}{6})

10 x - 3 \sqrt{3} y - \frac{5 π}{3} + 2 \sqrt{3} = 0

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